Biomedical Engineering Reference
In-Depth Information
either without the knowledge of the other. We will show that by using an iterative
algorithm based on fuzzy logic, we can estimate both.
9.4.4.2 Bias Corrected Fuzzy C-means (BCFCM)
Objective Function
Substituting Eq. 9.36 into Eq. 9.25, we have
y
r
∈
N
k
||
y
r
−
β
r
−
v
i
||
2
c
N
c
N
+
N
R
u
ik
||
y
k
−
β
k
−
v
i
||
u
ik
2
J
m
=
.
i
=
1
k
=
1
i
=
1
k
=
1
(9.37)
Formally, the optimization problem comes in the form
min
U
,
{
v
i
}
J
m
subject to
U
∈
U.
(9.38)
c
i
=
1
,
{
β
k
}
N
k
=
1
9.4.4.3 BCFCM Parameter Estimation
The objective function
J
m
can be minimized in a fashion similar to the MFCM
algorithm. Taking the first derivatives of
J
m
with respect to
u
ik
,
v
i
, and
β
k
and
setting them to zero results in three necessary but not sufficient conditions for
J
m
to be at a local extrema. In the following subsections, we will derive these
three conditions.
9.4.4.4 Membership Evaluation
Similar to the MFCM algorithm, the constrained optimization in Eq. 9.38 will be
solved using one Lagrange multiplier
u
ik
D
ik
+
N
R
u
ik
γ
i
c
N
c
F
m
=
1
−
u
ik
(9.39)
+
λ
i
=
1
k
=
1
i
=
1
y
r
∈
N
k
||
y
r
−
β
r
−
v
i
||
2
.
2
where
D
ik
=||
y
k
−
β
k
−
v
i
||
and
The
zero-
γ
i
=
gradient condition for the membership estimator can be written as
1
c
j
=
1
D
ik
+
N
R
γ
i
u
ik
=
p
−
1
.
(9.40)
1
D
jk
+
N
R
γ
j
